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(4.1b) lower approximation
(4.1c) upper approximation
FIGURE 4.1: Upper and lower approximation of a set A in the partitioned space of the
form (j, j + 1] 2
Topology of Rough sets
To study the topological properties of rough set, we define a partition topology on X, based
on partitions induced by the equivalence relation∼ X,ϕ . The equivalence classes x /∼ form
the basis for the topology τ (Steen and Seebach, 1995).
Let A denote the closure of a set A. Since the topology is a partition topology, to find
the closure A of a subset A⊂X, we have to consider all of the closed sets containing the
set A and then select the smallest closed set. The interior of a set A (denoted A) is the
largest open set that is contained in A. Open sets and their corresponding closed sets in
the topology are:
•X is an open set, φ is the corresponding closed set.
•φ is a closed set, X is the corresponding closed set.
•x i / ∼ is an open set⇒X\x i / ∼ = x i / ∼ , is a closed set.
where i = 1, . . . , n, and n is the total number of equivalence classes in the partition topology
and\is the set difference. Recall that the union of any finite or infinite number of open
sets are open sets and the intersection of any finite number of open sets is an open set. For
a set A⊆X and the partition topology τ , we have:
A = A −
A = A − (4.19)
where A, A,A − and A − are closure, interior, upper and lower approximation of A, respec-
tively. Now it is clear that we could define properties of a rough set in the language of
topology. Next, define π−boundary of A as: A b = A\A, where A and A are closure and
interior of A, (Lashin, Kozae, Abo Khadra, and Medhat, 2005). A set A is said to be rough,
if A b
6= φ, otherwise it is an exact set. Generally, for a given topology τ and A⊆X, we
A = A.
•A is totally definable, if A is an exact set, A =
•A is internally definable, if A = A, A 6= A.
•A is externally definable, if A 6= A, A = A.
•A is undefinable, if A 6=
A and A 6= A.